8.1 graphing f x ax2...parabola that opens down is the vertex. the vertex of the graph of f (x) = x2...

Section 8.1 Graphing f(x) = ax2 419

Graphing f (x) = ax28.1

Essential QuestionEssential Question What are some of the characteristics of the

graph of a quadratic function of the form f (x) = ax2?

Graphing Quadratic Functions

Work with a partner. Graph each quadratic function. Compare each graph to the

graph of f (x) = x2.

a. g(x) = 3x2 b. g(x) = −5x2

2

4

6

8

10

x2 4 6−6 −4 −2

y

f(x) = x2

4

x2 4 6−6 −4 −2

−4

−8

−12

−16

y

f(x) = x2

c. g(x) = −0.2x2 d. g(x) = 1 —

10 x2

2

4

6

x2 4 6−6 −4 −2

−2

−4

−6

y

f(x) = x2

2

4

6

8

10

x2 4 6−6 −4 −2

y

f(x) = x2

Communicate Your AnswerCommunicate Your Answer 2. What are some of the characteristics of the graph of a quadratic function of

the form f (x) = ax2?

3. How does the value of a affect the graph of f (x) = ax2? Consider 0 < a < 1,

a > 1, −1 < a < 0, and a < −1. Use a graphing calculator to verify

your answers.

4. The fi gure shows the graph of a quadratic function

of the form y = ax2. Which of the intervals

in Question 3 describes the value of a? Explain

your reasoning.

REASONING QUANTITATIVELY

To be profi cient in math, you need to make sense of quantities and their relationships in problem situations.

6

−1

−6

7

hsnb_alg1_pe_0801.indd 419hsnb_alg1_pe_0801.indd 419 2/5/15 8:40 AM2/5/15 8:40 AM

420 Chapter 8 Graphing Quadratic Functions

8.1 Lesson What You Will LearnWhat You Will Learn Identify characteristics of quadratic functions.

Graph and use quadratic functions of the form f (x) = ax2.

Identifying Characteristics of Quadratic FunctionsA quadratic function is a nonlinear function that can be written in the standard form

y = ax2 + bx + c, where a ≠ 0. The U-shaped graph of a quadratic function is called

a parabola. In this lesson, you will graph quadratic functions, where b and c equal 0.

Identifying Characteristics of a Quadratic Function

Consider the graph of the quadratic function.

Using the graph, you can identify characteristics such as the vertex, axis of symmetry,

and the behavior of the graph, as shown.

2

4

6

x42−4−6

y

axis ofsymmetry:x = −1

vertex:(−1, −2)

Function isdecreasing.

Function isincreasing.

You can also determine the following:

• The domain is all real numbers.

• The range is all real numbers greater than or equal to −2.

• When x < −1, y increases as x decreases.

• When x > −1, y increases as x increases.

quadratic function, p. 420parabola, p. 420vertex, p. 420axis of symmetry, p. 420

Previousdomainrangevertical shrinkvertical stretchrefl ection

Core VocabularyCore Vocabullarry

Core Core ConceptConceptCharacteristics of Quadratic FunctionsThe parent quadratic function is f (x) = x2. The graphs of all other quadratic

functions are transformations of the graph of the parent quadratic function.

The lowest point

on a parabola that

opens up or the

highest point on a

parabola that opens

down is the vertex.

The vertex of the

graph of f (x) = x2

is (0, 0).

x

y

axis ofsymmetry

vertex

decreasing increasing

The vertical line that

divides the parabola

into two symmetric

parts is the axis of symmetry. The axis

of symmetry passes

through the vertex. For

the graph of f (x) = x2,

the axis of symmetry

is the y-axis, or x = 0.

REMEMBERThe notation f (x) is another name for y.



Monitoring ProgressMonitoring Progress Help in English and Spanish at BigIdeasMath.com

Identify characteristics of the quadratic function and its graph.

1.

−2

−4

1

x642−2

y 2.

4

2

6

8

x1−3−7 −1

y

Graphing and Using f (x) = ax2

Core Core ConceptConceptGraphing f (x) = ax2 When a > 0• When 0 < a < 1, the graph of f (x) = ax2

is a vertical shrink of the graph of

f (x) = x2.

• When a > 1, the graph of f (x) = ax2 is a

vertical stretch of the graph of f (x) = x2.

Graphing f (x) = ax2 When a < 0• When −1 < a < 0, the graph of

f (x) = ax2 is a vertical shrink with a

refl ection in the x-axis of the graph

of f (x) = x2.

• When a < −1, the graph of f (x) = ax2

is a vertical stretch with a refl ection in

the x-axis of the graph of f (x) = x2.

REMEMBERThe graph of y = a ⋅ f (x) is a vertical stretch or shrink by a factor of a of the graph of y = f (x).

The graph of y = −f (x) is a refl ection in the x-axis of the graph of y = f (x).

Graphing y = ax2 When a > 0

Graph g(x) = 2x2. Compare the graph to the graph of f (x) = x2.

SOLUTION

Step 1 Make a table

of values.

Step 2 Plot the ordered pairs.

Step 3 Draw a smooth curve through the points.

Both graphs open up and have the same vertex, (0, 0), and the same axis of

symmetry, x = 0. The graph of g is narrower than the graph of f because the

graph of g is a vertical stretch by a factor of 2 of the graph of f.

x −2 −1 0 1 2

g(x) 8 2 0 2 8

x

a = 1 a > 1

0 < a < 1

y

x

a = −1 a < −1

−1 < a < 0

y

2

4

6

8

x42−2−4

y

g(x) = 2x2 f(x) = x2



Graphing y = ax2 When a < 0

Graph h(x) = − 1 — 3 x2. Compare the graph to the graph of f (x) = x2.

SOLUTION

Step 1 Make a table of values.

x −6 −3 0 3 6

h(x) −12 −3 0 −3 −12

Step 2 Plot the ordered pairs.

Step 3 Draw a smooth curve through the points.

The graphs have the same vertex, (0, 0),

and the same axis of symmetry, x = 0, but

the graph of h opens down and is wider

than the graph of f. So, the graph of h is a

vertical shrink by a factor of 1 —

3 and a

refl ection in the x-axis of the graph of f.


Graph the function. Compare the graph to the graph of f (x) = x2.

3. g(x) = 5x2 4. h(x) = 1 —

3 x2 5. n(x) =

3 —

2 x2

6. p(x) = −3x2 7. q(x) = −0.1x2 8. g(x) = − 1 — 4 x2

Solving a Real-Life Problem

The diagram at the left shows the cross section of a satellite dish, where x and y are

measured in meters. Find the width and depth of the dish.

SOLUTION

Use the domain of the function to fi nd the width

of the dish. Use the range to fi nd the depth.

The leftmost point on the graph is (−2, 1), and

the rightmost point is (2, 1). So, the domain

is −2 ≤ x ≤ 2, which represents 4 meters.

The lowest point on the graph is (0, 0), and the highest points on the graph

are (−2, 1) and (2, 1). So, the range is 0 ≤ y ≤ 1, which represents 1 meter.

So, the satellite dish is 4 meters wide and 1 meter deep.


9. The cross section of a spotlight can be modeled by the graph of y = 0.5x2,

where x and y are measured in inches and −2 ≤ x ≤ 2. Find the width and

depth of the spotlight.

STUDY TIPTo make the calculations easier, choose x-values that are multiples of 3.

−4

−8

−12

4

x84−4−8

y

f(x) = x2

h(x) = − x213

1

2

x2−1−2

y

y = x214(−2, 1)

(0, 0)

(2, 1)

width

depth



Exercises8.1 Dynamic Solutions available at BigIdeasMath.com

In Exercises 3 and 4, identify characteristics of the quadratic function and its graph. (See Example 1.)

3.

−2

−6

1

x42−2

y

4.

4

8

12

x84−4−8

y

In Exercises 5–12, graph the function. Compare the graph to the graph of f (x) = x2. (See Examples 2 and 3.)

5. g(x) = 6x2 6. b(x) = 2.5x2

7. h(x) = 1 — 4 x2 8. j(x) = 0.75x2

9. m(x) = −2x2 10. q(x) = − 9 — 2 x2

11. k(x) = −0.2x2 12. p(x) = − 2 — 3 x2

In Exercises 13–16, use a graphing calculator to graph the function. Compare the graph to the graph of y = −4x2.

13. y = 4x2 14. y = −0.4x2

15. y = −0.04x2 16. y = −0.004x2

17. ERROR ANALYSIS Describe and correct the error in

graphing and comparing y = x2 and y = 0.5x2.

x

y = x2

y = 0.5x2

y

The graphs have the same vertex and the same

axis of symmetry. The graph of y = 0.5x2 is

narrower than the graph of y = x2.

✗

18. MODELING WITH MATHEMATICS The arch support of

a bridge can be modeled by y = −0.0012x2, where x

and y are measured in feet. Find the height and width

of the arch. (See Example 4.)

50

x350 45025050−50−250−350−450

y

−150

−250

−350

19. PROBLEM SOLVING The breaking strength z

(in pounds) of a manila rope can be modeled by

z = 8900d2, where d is the diameter (in inches)

of the rope.

a. Describe the domain and

range of the function.

b. Graph the function using

the domain in part (a).

c. A manila rope has four times

the breaking strength of another manila rope. Does

the stronger rope have four times the diameter?

Explain.

Monitoring Progress and Modeling with MathematicsMonitoring Progress and Modeling with Mathematics

1. VOCABULARY What is the U-shaped graph of a quadratic function called?

2. WRITING When does the graph of a quadratic function open up? open down?

Vocabulary and Core Concept CheckVocabulary and Core Concept Check

d



20. HOW DO YOU SEE IT? Describe the possible values

of a.

a.

x

y

g(x) = ax2 f(x) = x2

b.

x

y

f(x) = x2

g(x) = ax2

ANALYZING GRAPHS In Exercises 21–23, use the graph.

x

f(x) = ax2, a > 0

g(x) = ax2, a < 0

y

21. When is each function increasing?

22. When is each function decreasing?

23. Which function could include the point (−2, 3)? Find

the value of a when the graph passes through (−2, 3).

24. REASONING Is the x-intercept of the graph of y = ax2

always 0? Justify your answer.

25. REASONING A parabola opens up and passes through

(−4, 2) and (6, −3). How do you know that (−4, 2) is

not the vertex?

ABSTRACT REASONING In Exercises 26–29, determine whether the statement is always, sometimes, or never true. Explain your reasoning.

26. The graph of f (x) = ax2 is narrower than the graph of

g(x) = x2 when a > 0.

27. The graph of f (x) = ax2 is narrower than the graph of

g(x) = x2 when ∣ a ∣ > 1.

28. The graph of f (x) = ax2 is wider than the graph of

g(x) = x2 when 0 < ∣ a ∣ < 1.

29. The graph of f (x) = ax2 is wider than the graph of

g(x) = dx2 when ∣ a ∣ > ∣ d ∣ .

30. THOUGHT PROVOKING Draw the isosceles triangle

shown. Divide each leg into eight congruent segments.

Connect the highest point of one leg with the lowest

point of the other leg. Then connect the second

highest point of one leg to the second lowest point of

the other leg. Continue this process. Write a quadratic

equation whose graph models the shape that appears.

base

legleg

(−6, −4) (6, −4)

(0, 4)

31. MAKING AN ARGUMENT The diagram shows the

parabolic cross section

of a swirling glass of

water, where x and y are

measured in centimeters.

a. About how wide is the

mouth of the glass?

b. Your friend claims that

the rotational speed of

the water would have

to increase for the

cross section to be

modeled by y = 0.1x2.

Is your friend correct?

Explain your reasoning.

Maintaining Mathematical ProficiencyMaintaining Mathematical ProficiencyEvaluate the expression when n = 3 and x = −2. (Skills Review Handbook)

32. n2 + 5 33. 3x2 − 9 34. −4n2 + 11 35. n + 2x2

Reviewing what you learned in previous grades and lessons

x

y(−4, 3.2) (4, 3.2)

(0, 0)


8.1 graphing f x ax2...parabola that opens down is the vertex. the vertex of the graph of f (x) = x2...

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